Elastic Constants of and Wave Propagation in Antimony and Bismuth

Abstract

Ultrasonic wave velocities for 14 different modes were obtained on two differently oriented single-crystal antimony cubes from the time between successive unrectified radio-frequency pulse echoes. This redundant set of data was fitted by a least-squares technique to Voigt theory to yield the six room-temperature adiabatic elastic-stiffness constants. In units of ${10}^{10}$ dyn/${\mathrm{cm}}^{2}$, ${c}_{11}=99.4(1)$, ${c}_{33}=44.5(9)$, ${c}_{44}=39.5(5)$, ${c}_{66}=34.2(3)$, ${c}_{13}=26.4(4)$, and ${c}_{14}=+21.6(4)$, the positive sign for ${c}_{14}$ following from our choice of positive Cartesian axes. When similarly treated, Eckstein, Lawson, and Reneker's bismuth data yield in these same units: ${c}_{11}=63.22$, ${c}_{33}=38.11$, ${c}_{44}=11.30$, ${c}_{66}=19.40$, ${c}_{13}=24.40\ifmmode\pm\else\textpm\fi{}0.09$, ${c}_{14}=+7.20$. Also included are a visual method of fixing the laboratory coordinate system in antimony by means of an imperfect cleavage plane, a calculation of the pure-mode directions in the mirror plane, a simple formula for choosing the nonextraneous value of ${c}_{13}$ for trigonal crystals having six independent elastic constants without resorting to lattice-stability criteria, and a calculation of the deviation of elastic-wave particle displacement and energy-flux directions from the propagation direction. For waves propagating in the (0,1,1) and ($0,\overline{1},1$) directions, the particle-displacement deviations for antimony and bismuth do not exceed 15\ifmmode^\circ\else\textdegree\fi{} and 13\ifmmode^\circ\else\textdegree\fi{}, respectively, and corresponding energy-flux deviations up to 45\ifmmode^\circ\else\textdegree\fi{} and 27\ifmmode^\circ\else\textdegree\fi{} are obtained.